Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 17F / ii

For $r\ge 3$, join a clique on $r-2$ vertices to an independent set on $n-r+2$ vertices. The resulting complete $(r-1)$-partite graph hasIts largest clique has size $r-1$. Every missing edge joins two vertices of the independent set; adding it produces a $K_r$ with the original $(r-2)$-clique. It is therefore maximal $K_r$-free. For $n>r$ its part sizes are unbalanced, so its edge count is strictly less than that of $T_{r-1}(n)$.

For $r=3$ the construction is a star with $n-1$ edges. Every maximal triangle-free graph on $n>3$ vertices is connected: an edge between distinct components would create no triangle. A connected graph has at least $n-1$ edges, so the minimum is $n-1$, attained by the star. Literally the requested construction is impossible for $r=2$, since $T_1(n)$ has zero edges. The nontrivial assertion thus requires the customary $r\ge 3$ hypothesis.